Symmetries of Convex Sets in the Hyperbolic Plane

We establish some results characterizing central or axial symmetry of convex sets in the hyperbolic plane. The characterizations follow the spirit of a Chakerian-Klamkin's characterization of central symmetry for Euclidean sets: if for any three point-subset M of a compact set K there is a symmetric image of M that is also contained in K, then K has a center of hyperbolic symmetry. We also study axial symmetry when the axis is either a geodesic, a horocycle, or a hypercycle. Finally, in the last section we give a characterization of the hyperbolic disc.